Journal of Mechanical Science and Technology 23 (2009) 1354~1364

www.springerlink.com/content/1738-494x DOI 10.1007/s12206-009-0350-3

Journal of

Mechanical Science and Technology

Variable structure fuzzy control using three input variables

for reducing motion tracking errors†

Chul-Goo Kang* Department of Mechanical Engineering, Konkuk University, Seoul, 143-701, Korea

(Manuscript Received February 21, 2007; Revised November 18, 2008; Accepted January 30, 2009)

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Abstract Conventional fuzzy controllers for motion tracking utilize generally two input variables (position error and velocity

error) to deal with highly nonlinear and time-varying dynamics associated with complex mechanical motion with mul-ti-DOF. This results in some tracking errors at steady state, in general, mainly due to friction existing in mechanical systems. To eliminate the steady-state tracking errors, a variable structure fuzzy control algorithm is proposed using three input variables (position error, velocity error, and integral of position errors) and a switching logic between two inputs and three inputs. Simulation and experimental studies have been conducted to show the validity of the proposed control logic using a direct-drive SCARA manipulator with two degree-of-freedom. It has been shown that the pro-posed fuzzy control logic has significantly improved motion-tracking performance of the mechanical system when it is applied to complex polygon-tracking in Cartesian space with inverse kinematics and path planning.

Keywords: Cartesian space; Fuzzy control; Motion tracking; Steady-state error; Variable structure control

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1. Introduction

Since the dynamics of complex mechanical sys-

tems such as robotic manipulators is highly nonlinear

and time-varying, a fast and accurate motion control

is a challenging problem [1]. The control by way of

exact cancellation of all dynamic disturbances such as

varying inertia, Coriolis and centrifugal accelerations,

and gravitational and frictional disturbances, or the

control by way of dealing with all dynamic distur-

bances as uncertainties, could be very complex and

sometimes may not be computationally feasible for

fast operating mechanical systems. In this case, fuzzy

control is a good candidate since it does not depend

on the accurate model of the plant, but is based on

heuristics about plant behaviors. Recently, various

fuzzy control logics have been tried to be applied to

motion control of mechanical systems [2-4].

Fuzzy set theory was first introduced by L. A.

Zadeh [5] to deal with imprecise objects. He sug-

gested various possible application fields of fuzzy set

theory (by relying on the use of linguistic variables

and fuzzy algorithms) where the behaviors of systems

are too complex or too ill-defined to admit of precise

mathematical analysis [6]. Mamdani [7] first success-

fully applied fuzzy logic algorithms for the control of

a small laboratory steam engine. Thereafter, many

applications of fuzzy control were reported in the

various engineering fields including chemical proc-

esses and consumer products [8-10]. A good sum-

mary of the fuzzy logic controls is presented by Lee

[11]. Wang [12] viewed fuzzy control theory as a

subset of nonlinear control theory. He pointed out

better control performance of fuzzy controllers com-

pared to conventional PID controllers is due to the

nonlinear behavior of fuzzy controllers.

Motion control problems require faster and more

accurate response compared with other industrial

†This paper was recommended for publication in revised form by Associate

Editor Kyongsu Yi

*Corresponding author. Tel.: +82 2 447 2142, Fax.: +82 2 447 5886

E-mail address: cgkang@konkuk.ac.kr

© KSME & Springer 2009

C.-G. Kang / Journal of Mechanical Science and Technology 23 (2009) 1354~1364 1355

processes. The applications of fuzzy control algo-

rithms to motion control problems are reported exten-

sively in the literature. Among them, Li and Lau [13]

applied fuzzy control to servomotor systems and

showed that the performance of the fuzzy controller is

better than that of PI controllers in terms of steady

state error, settling time and response time through

simulation study using two look-up tables for coarse

and fine controls. Huang and Tomizuka [14] applied

the fuzzy controller to two-dimensional motion track-

ing control and showed tracking precision and travel

time can be improved compared with the system with

conventional PD controller by simulation study.

Scharf and Mandic [15] applied it to motion control

of an indirect drive robot manipulator and showed

that step response and tracking performance are often

superior to those of conventional PID controllers.

However, the effect of time-varying and nonlinear

dynamics in indirect drive is not so severe. Kang et

al.[16] applied fuzzy control to motion tracking con-

trol of direct drive robotic manipulators, and showed

that position tracking performance of fuzzy control is

similar to or often better than the one of PID control

through simulation and experimental studies.

Recently, Kohn-Rich et al.[2], and Ham et al.[3]

designed stable fuzzy controllers for a large class of

mechanical systems in the framework of Lyapunov’s

stability theory. Park et al.[4] proposed a sliding

mode control logic with fuzzy adaptive perturbation

compensator for parallel manipulators. Adams and

Rattan [17] adopted a multi-stage fuzzy PID control

scheme with fuzzy switching to reduce steady-state

errors. Adams and Rattan’s solution shows a nice way

to include fuzzy I action in the controller, in which

fuzzy PD control output is blended with fuzzy I ac-

tion before defuzzifying them when fuzzy PD input is

in the Zero fuzzy set. Sooraksa et al.[18] applied their

fuzzy P2ID scheme to handlebar control of a bicycle

robot.

In this paper, a variable structure fuzzy control

logic, a revised algorithm of Kang et al.’s one [16,19],

is proposed using three input variables of fuzzy con-

troller (position error, velocity error and integral of

position errors) and a switching logic between two

inputs and three inputs in order to eliminate position

errors existing at steady state in motion control, and it

is applied to motion control of a mechanical manipu-

lator.

The present paper has extended the result of the au-

thor’s previous publication [20] to motion control of

Cartesian space additionally using inverse kinematics

and Jacobian of the manipulator. The proposed con-

trol scheme is different from Adams and Rattan’s

solution [17] in the sense that two fuzzy control logics,

fuzzy PD and fuzzy PID control logic, are designed

separately (not multi-stage), and the switching be-

tween fuzzy PD and fuzzy PID control occurs accord-

ing to tracking condition, that is, transient or steady-

state condition. The idea of the proposed control

scheme is relatively simple but the improvement of

actual control performance is significant. This fact has

been shown through simulation and experimental

studies using a direct drive SCARA manipulator with

two degree-of-freedom (DOF) motion.

The paper is organized as follows. Section 2 pre-

sents basic terminologies and the structure of the pro-

posed fuzzy control scheme, and Section 3 describes

application of the proposed fuzzy control scheme to

motion tracking of a mechanical system. In Section 4,

computer simulation and experimental results of the

proposed fuzzy control are discussed, and concluding remarks are given in Section 5.

2. Variable structure fuzzy control

2.1 Definitions and terminology

A fuzzy set A in X is characterized by a membership

function ( )xm which associates with each point x in

X a real number in the interval [0, 1], with the value 0

representing non-membership and the value 1 repre-

senting full membership. The support of a fuzzy set A

is the crisp set of all points x in X such that ( ) 0xm > .

A fuzzy set whose support is a single point in X with

( )xm =1 is referred to as fuzzy singleton. Fuzzy set

operations and fuzzy logic can be defined in many

ways [21]. In the following, we summarize the defini-

tions and terminology used in this paper. The mem-

bership function ( )xm of the union A BÈ of two

fuzzy sets A and B is pointwise defined for all x in X

by ( )A B xm È = max{ ( ), ( )},A Bx x x Xm m Î , and the

membership function ( )xm of the intersection

A BÇ is defined for all x in X by ( )A B xm Ç =

min{ ( ), ( )},A Bx x x Xm m Î , which are originally

proposed by Zadeh [5]. Fuzzy implication adopted

in this paper is Larsen's product rule [12] among

many other choices, which is defined by ( , )A B x ym ®

( ) ( )A Bx ym m= , ,x X y YÎ Î . Fuzzy reasoning is

generally based on a compositional rule of inference

[6], which can be viewed as an extension of the mo-

dus ponens. The following sup-min composition pro-

1356 C.-G. Kang / Journal of Mechanical Science and Technology 23 (2009) 1354~1364

posed by Zadeh is used in this paper.

( )

( ) sup min{ ( ), ( ) ( )}B A A Bx X

B A A B

y x x ym m m m¢ ¢Î

¢ ¢= ®

=

o

A linguistic variable is a variable that can take

words in natural languages (for example, big, small

etc.) as its values. These words are usually labels of

fuzzy sets.

2.2 Proposed variable structure fuzzy controller

Position errors due to Coulomb friction, backlash

etc., generally remain at steady state in position con-

trol of mechanical systems such as robotic manipula-

tors. A conventional fuzzy PD controller using two

input variables (position error and velocity error) [16]

cannot remove such errors existing at steady state,

e.g., at a designation point of the end-effector of a

robotic manipulator.

From the idea that steady state errors can be gener-

ally reduced if we add an integral action to propor-

tional and derivative control, conventional fuzzy PD

controller with two input variables is revised to a

fuzzy controller with variable structure between fuzzy

PD control and fuzzy PID control logic with three

input variables – position error, velocity error, and

integral of position errors – and with a switching logic

to eliminate position errors existing at steady state in

motion control of mechanical systems. Since position

errors at steady state are usually small, we integrate

position errors only during steady state and use inte-

gral values as another input variable of the fuzzy con-

troller to remove small position errors effectively.

The proposed algorithm includes two types of

fuzzy control logic: fuzzy PD control and fuzzy PID

control logic. If motion tracking is in transient mode,

the fuzzy PD control logic is activated, and if it is in a

steady-state mode, the fuzzy PID control logic is acti-

vated. The switching between fuzzy PD control and

fuzzy PID control occurs in a crisp way in this paper

for simplicity, although the decision related to the

steady state detection could be made in a fuzzy way.

That is, if the change of position error during one

sampling period is below a threshold value and the

position error is below the preset maximum steady

state error, then the flag for error integral is turned on

and the fuzzy controller with two input variables is

switched to the one with three input variables.

If the change of position error during one sampling

period becomes again equal to or greater than the

threshold value, or the position error becomes equal

to or greater than the preset maximum steady state

error, then the flag for error integral is turned off and

the fuzzy controller with three input variables is re-

stored to the previous fuzzy controller with two input

variables.

Even though the stability of the fuzzy logic control

system is an important issue, we do not consider the

stability problem here and assume the overall system

is heuristically stable. For stability issues of fuzzy

control systems, refer to the reference papers [2, 3, 18,

22].

A block diagram of the proposed variable structure

fuzzy control system is shown in Fig. 1. The sampler

and zero-order hold (ZOH) are implemented by timer

interrupt and D/A converter (DAC). The position and

velocity signals are digital and the fuzzy control ac-

tion is implemented by using a microprocessor and

random access memory (for a look-up table).

The input variables of the fuzzy controller are an-

gular position error e (rad) and angular velocity error

e& (rad/s), and integral of angular position errors eò

(rad × s). The output variable of the fuzzy controller is

the control input u (N ×m) to the motor driver.

( )

,

( ) ( ( 1) ),

( , ) ( , , )

k

i j

e position command actual position

e kT e k Te

T

e e iT T

u f e e or u f e e e

=

= -

- -=

=

= =

åò

ò

&

& &

where T represents sampling time, and i, j and k are

positive integers such that k represents present time

index and j represents time index at which the steady

state condition is detected. In the fuzzy controller, the

measured e, e& and eò values are scaled to some

real values in the interval [-1, 1] and mapped to lin-

guistic variables E, DE and SE by the fuzzification

operator. The values of linguistic variables are com-

posed of linguistic terms PL, PS, ZO, NS and NL, or

-ZOH Plant

uActual

position

+

dt

d

e

e

Switching

Fuzzy

controller

Positioncommand

òò

e

•

• •

Fig. 1. Block diagram of the proposed fuzzy control system.

C.-G. Kang / Journal of Mechanical Science and Technology 23 (2009) 1354~1364 1357

ZONS PSNL PL

-1.0 0 1.0-0.5 0.5

m

(a)

Z PN

-1.0 0.0 1.0-0.5 0.5

0.5

1.0

(b) Fig. 2. Membership functions of fuzzy sets for (a) position error, velocity error, control input, and (b) integral of position errors.

N, Z and P that are all fuzzy sets.

This linguistic term set forms a fuzzy partition of

input and output spaces. The knowledge base of the

fuzzy controller is composed of a database and a rule

base. The database defines membership functions for

the above linguistic terms, and the rule base repre-

sents fuzzy control rules. Instead of quantizing the

universe of discourse of input and output spaces into a

finite number of segments (quantization levels), we

do a linear scale mapping of position error, velocity

error, integral of position errors and control input with

a value in [-1, 1], and so keep infinite resolutions in

the fuzzification and defuzzification operations (that

is, we do not lose some information from the given

data at these stages).

Membership functions of fuzzy sets (linguistic

terms) and fuzzy control rules have a considerable

effect on control performance. Fig. 2 shows the

membership functions of the fuzzy sets for position

error e, velocity error e& , control input u, and integral

of position errors eò .

We use membership functions with the same trian-

gular shapes and same support sizes, such as Fig. 2.

However, we also can use membership functions with

the same triangular shapes but different support sizes.

With membership functions with different support

sizes, we can emphasize that the control near zero is

more important than that of the other portions.

A fuzzy control rule is a fuzzy conditional statement

(IF-THEN statement) in which the antecedents and

the consequent are associated with fuzzy concepts

(linguistic terms). One example of a fuzzy control

rule is as follows: R1: If E is ZO and DE is ZO and SE is N, then U is NS.

R2: If E is PS and DE is NS and SE is Z, then U is ZO.

m

1

0 0.5DE

m

1

0-1SE

m

1

0-1DE

m

1

0 1

m

1

0 1E

m

1

0 0.5E

m

1

0U

-1

SE-0.25-0.250.3

m

1

0 0.5U

m

1

0 0.5U

-1

0.25

0.5

R1

R2

Fig. 3. Graphical representation of the inference mechanism.

These rules are implemented by fuzzy relation R1 =

{(ZO and ZO and N) ® NS} and R2 = {(PS and NS

and Z) ® ZO} where 'and' and '® ' are processed

by min operation (intersection) and Larsen's product

rule, respectively.

Fuzzy reasoning (or approximate reasoning) is

done by the sup-min composition operator. As an

illustration, consider the above two fuzzy control

rules R1 and R2. Let the inputs be fuzzy singletons,

namely, E’ = 0.3, DE’ = -0.25 and SE’ = -0.25. Then

the inference mechanism adopted here can be repre-

sented graphically as shown in Fig. 3.

Defuzzification is a mapping from fuzzy control ac-

tions into nonfuzzy (crisp) control actions. Among

many defuzzification strategies, we adopt the center

average method [12].

The calculation time consumed for fuzzy inference

and fuzzification is usually considerable (e.g., a few

milliseconds). To save the calculation time in real-

time control, we calculate U for the standard E, DE

and SE values in advance, and form a three-

dimensional look-up table. Since the number of input

variables could be two or three according to tracking

situations (transient or steady state), we need two-

dimensional and three dimensional look-up tables

together. Roughly speaking, a two-dimensional look-

up table is necessary for transient condition, and a

three-dimensional look-up table is for steady state

condition. These two look-up tables are used alter-

nately as time passes.

3. Implementation of the proposed control scheme to motion tracking

The mechanical system with two DOF motion con-

sidered in this paper is shown in Fig. 4. This system is

actually a two axes SCARA manipulator moving in a

horizontal plane. The control system has two inputs

and four outputs. Two inputs are two motor-driving

signals, and four outputs are two position signals and

two velocity signals of two motors coming from re-

solvers. Furthermore, this mechanical system has

highly nonlinear and time-varying dynamics including

1358 C.-G. Kang / Journal of Mechanical Science and Technology 23 (2009) 1354~1364

Fig. 4. Photograph of the 2 DOF mechanical system used in

simulation and experimental studies.

Coriolis acceleration and coupling effects between

axes. This manipulator is driven by two direct drive

motors (Megatorque motors). This system was de-

signed and integrated in our laboratory. The range of

input voltages to driver units is -10 volts to 10 volts.

Torque generated by the Megatorque motor is pro-

portional to input voltage. Maximum torques of the

base motor and the upper motor are 147 N�m and 9.8

N�m, respectively. The resolver is attached to each

Megatorque motor to measure angular position and

velocity. Resolver signals are converted to phase A

and B quadrature signals by RDC (Resolver to Digital

Converter) in the Driver Unit, and these quadrature

signals are counted by each counter board. The reso-

lutions of feedback signals of the base motor and the

upper motor are 38,400 counts/rev and 25,600

counts/rev, respectively. Maximum speeds of the base

motor and the upper motor are 3 rev/sec and 4.5

rev/sec, respectively.

A dynamic model of the robot is not necessary for

designing a fuzzy controller, but we develop one for

simulation purpose. The dynamics of an n-DOF, rigid

link, direct drive manipulator can be generally de-

scribed as follows [23, 24]:

( ( )) ( ) ( ( ), ( )) ( ( )) ( ( ), ( ))

( )

t t t t t t t

t

q w q w q w t

t

+ + +

=

M v g f&

where q is an angular position vector of joints, w

is an angular velocity vector of joints, t is an input

torque vector supplied by actuators, M is a symmet-

ric and positive definite generalized inertia matrix, v

is a vector due to Coriolis and centrifugal forces, f is

viscous and Coulomb friction torque vector, and g

is a gravitational torque vector. For the two-axes di-

rect-drive manipulator considered in this paper, each

term in the above equation is given as follows:

Table 1. Scale mapping of the position error, velocity error,

integral of position errors and control input.

Scaled

values 1e

rad

2e

rad

1eò

rad�s

2eò

rad�s

1 2,e e& &

rad/s

1u

N�m

2u

N�m

-1.0 -0.30 -0.10 -3.0e-4 -8.0e-4 -4.0 -145 -9.8

-0.5 -0.15 -0.05 -1.5e-4 -4.0e-4 -2.0 -72.5 -4.9

0 0 0 0 0 0 0 0

0.5 0.15 0.05 1.5e-4 4.0e-4 2.0 72.5 4.9

1.0 0.30 0.10 3.0e-4 8.0e-4 4.0 145 9.8

1 11

2 2 2

( ) ( )( )( ) , ( ) , ( )

( ) ( ) ( )

t ttt t t

t t t

w tqq w t

q w t

é ù é ùé ù= = =ê ú ê úê úë û ë û ë û

11 3 1 211 12

12 2 1 212 22

22 2

2 cos

( ) , cos

m p pm m

m p pm m

m p

q

q q

= +é ù

= = +ê úë û =

M

2 1 2 1 2

21 1 2

(2 ) sin( , )

sin

p

p

w w w qq w

w q

- +é ù= ê úê úë û

v

1 1

2 2

0.24( , ) ,

0.035

sign if 0

sign if 0 and

if 0 and

i i i

i i i i i i

i i i i

f

f

T

f T T

T

ww t

w

w w

t w t

t w t

´ +é ù= ê ú

´ +ë û

ì >ï

= = >íï = £î

f

1 23.2 , 0.17T N m T N m= × = ×

( ) 0q =g

where

1

2

3

2

0.123 0.08

0.138 0.0625

1.676 0.165

0 3.76

0 0.012

p

p p

p p

p

p

p M

p M I

p M I

M or kg

I or kg m

= +

= + +

= + +

=

= ×

pM is the mass of payload and pI is the mo-

ment of inertia of payload. All the values above are

represented in SI units.

For this mechanical system, the fuzzy controller

proposed in the previous section is applied. First we

define a linear scale mapping for position error, ve-

locity error, integral of position errors and control

input by values in [-1, 1]. Table 1 shows a summary

of the linear scale mapping where subscripts 1 and 2

represent the base motor and the upper motor, respec-

tively.

C.-G. Kang / Journal of Mechanical Science and Technology 23 (2009) 1354~1364 1359

In the table, the maximum value of the control in-

put is the same with the maximum torque that Mega-

torque motor can generate. These maximum values

are the values that we allow at the corresponding

variables. In some sense, these values act like the

inverses of the proportional and derivative gain in PD

control. However, the difference of the fuzzy control-

ler and PD controller is that the fuzzy controller acts

in a nonlinear fashion.

The whole fuzzy control rules of the fuzzy control-

ler for this mechanical system are shown in Table 2.

Linguistic terms in Table 2 are defined in Fig. 2.

In Table 2, the integral term (SE) is activated (i.e.,

summed) only when the steady state condition is de-

tected from the way proposed in Section 2.2 in which

the flag for error integral is used. At the steady state

condition, position error (E) and velocity error (DE)

are both ZO. In this way, the number of fuzzy control

rules is just increased from 25 to 27 even if the num-

ber of input variables is increased from 2 to 3.

After fuzzy inference and defuzzification processes

presented in the previous section, the actual control

input u is obtained by multiplying the scaling factor

145 (for base motor) or 9.8 (for upper motor). Ac-

cording to control input u, actual torque t is gener-

ated by the motor.

For the present mechanical system, one three-

dimensional look-up table instead of two look-up

tables can be generated and used from the given fuzzy

control rules in Table 2. In view of the required

memory size and accuracy of the fuzzy inference,

21x21 U values in the look-up table are generated for

SE = 0, and 11x11x21 U values are generated for the

other SE values. For SE = 0, the values of E and DE

are divided into interval of 0.1 from –1 to 1. For the

other SE values, the values of E and DE are divided

into the same interval from -0.5 to 0.5, and SE into

the same interval from –1 to 1.

In this way, we save the required memory size up

to 70 % compared to the memory size required for

three-dimensional look-up table (21x21x21) with

intervals of 0.1 for three input variables. The 21x21

table for SE = 0 can be considered as the two-

dimensional look-up table that is necessary for a tran-

sient condition of motion tracking. For a specific po-

sition error, velocity error and integral of position

errors, control input is calculated through a linear

interpolation by using look-up tables as shown in Fig.

5. For specific input values, the interpolated fuzzy

controller output U is given as follows (using the

Table 2. Fuzzy control rules for position control of the robotic

manipulator.

If input variables are E and DE (transient condition), or if in-

put variables are E, DE and SE with SE=Z (steady-state condi-

tion), then U are

\E DE NL NS ZO PS PL

NL NL NL NL NS ZO

NS NL NL NS ZO PS

ZO NL NS ZO PS PL

PS NS ZO PS PL PL

PL ZO PS PL PL PL

If E = ZO, DE = ZO and SE = P (steady-state condition),

then U = PS.

If E = ZO, DE = ZO and SE = N (steady-state condition),

then U = NS.

Fig. 5. Interpolation in the three dimensional lookup table.

notations in Fig. 5):

1 2

3 4 5

6 7 8

1 1 1 1 1

1 1 1

1 1 1 1

x y z x y zU U U

x y z x y z

x y z x y z x y zU U U

x yz x y z xyz

x y z x y z x z yU U U

x y z xy z xz y

æ ö æ öD D D D D Dæ ö æ ö æ ö= - - - + - -ç ÷ ç ÷ç ÷ ç ÷ ç ÷è ø è ø è øè ø è ø

D D D D D D D D Dæ ö æ ö æ ö+ - + - - +ç ÷ ç ÷ ç ÷è ø è ø è ø

æ ö æ öD D D D D D D D Dæ ö æ ö+ - - + - + -ç ÷ ç ÷ç ÷ ç ÷

è ø è øè ø è ø

As the tracking condition (transient or steady-state)

of the manipulator is changed, the proposed control

scheme switches the fuzzy PD control logic to the

fuzzy PID control logic and vice versa according to

the switching logic explained in the previous section.

4. Simulation and experimental results

For the 2 DOF mechanical system described in the

previous section, digital simulation and experimental

studies have been conducted to show that the pro-

posed variable structure fuzzy controller reduces

1360 C.-G. Kang / Journal of Mechanical Science and Technology 23 (2009) 1354~1364

steady-state position errors significantly compared to

conventional fuzzy controllers. One simple chosen

task is to drive each axis of the manipulator inde-

pendently in joint space without considering total

end-effector motion, which was shown in the author’s

previous Korean publication [20]. Here the results are

summarized briefly again for comparison purposes.

For this task, the desired trajectory for both axes is

given by the seventh-order polynomial as shown in

Fig. 6. That is, the task is to drive both motors at the

same time with the trajectory given in Fig. 6. The

overlapped curve with the desired trajectory in Fig. 6

is an experimental result for the upper axis when the

proposed fuzzy control is applied for this task.

Since the distinction between the desired trajectory

and the actual trajectory is difficult in Fig. 6, we plot

position errors in a magnified scale to evaluate con-

trol performance clearly. Fig. 7 (a) shows the simula-

tion result, and Fig. 7 (b) shows the experimental

result for position error of the base axis when the

proposed fuzzy controller (denoted by “Proposed

Fuzzy” in the figures) and the conventional fuzzy

controller (denoted by “Old Fuzzy”) are applied with

3.76 kg payload. Here the conventional fuzzy control-

ler implies the fuzzy controller with two input vari-

ables. From Fig. 7 (a) and Fig. 7 (b), we see that the

proposed fuzzy control reduces position errors more

than 80 % (in magnitude) at steady state compared to

those of the conventional fuzzy control, and has simi-

lar position tracking errors to those of the conven-

tional fuzzy control at transient response intervals.

The simulation and experimental results for the

steady state position errors are summarized in Table 3.

Fig. 6. Desired and actual position trajectories of the upper motor when the proposed fuzzy control is applied without pay-

load (experiment).

Table 3. Simulation and experiment results for the steady state

position errors.

Convent.

fuzzy (A)

Proposed

fuzzy (B) B/A

Simul. Base axis

(no payload) -2.5e-3 rad 0.5e-4 rad 2 %

Upper axis

(no payload) -2.0e-3 rad 1.6e-4 rad 8 %

Base axis

(3.76 kg ) 5.4e-3 rad -1.1e-4 rad 2 %

Upper axis

(3.76 kg ) 0.3e-3 rad 0.3e-4 rad 10 %

Exper. Base axis

(no payload) -9.8e-3 rad 6.8e-4 rad 7 %

Upper axis

(no payload) 2.0e-3 rad 3.0e-4 rad 15 %

Base axis (3.76 kg )

-10.8e-3 rad 1.1e-4 rad 1 %

Upper axis

(3.76 kg ) 1.2e-3 rad 2.0e-4 rad 17 %

(a)

(b) Fig. 7. Position tracking errors of the base axis when the pro-

posed and the conventional fuzzy control are applied with 3.76

kg payload. (a) simulation, (b) experiment.

C.-G. Kang / Journal of Mechanical Science and Technology 23 (2009) 1354~1364 1361

However, the motion of the manipulator should be

controlled in Cartesian space in order to be applied to

actual factory operations. Thus, the previous results

are extended to a complex task to drive the manipula-

tor in Cartesian space using inverse kinematics and

path planning. The selected Cartesian space task is to

draw various polygons on the table with the manipu-

lator.

In the simulation, the controller is realized in a dis-

crete time domain, but the manipulator is realized as a

continuous time model (using a 4th-order Runge-

Kutta algorithm). Sampling time T is set to 5 ms in

the simulation and experiment. In the experiment,

elapsed time for executing the interrupt service rou-

tine within one sampling interval was measured to be

about 0.3 ms. To keep sampling intervals exact in the

experiments, we used timer interrupts. Programming

for simulations and experiments was coded with C

language.

To realize tracking control in Cartesian space, we

have to solve the inverse kinematics problem on-line.

The solution of the inverse kinematics problem and

manipulator Jacobian are obtained as closed-forms for

this SCARA manipulator as follows:

2 2 2 2 2 2 21 2 1 21

2 2 2 2 21 2 1 2

1 1 2 21

1 2 2

1 1 2 1 2 2 1 2

1 1 2 1 2 2 1 2

1 ( ) /(4 )tan

( ) /(2 )

sintan tan

cos

sin sin( ), sin( )

cos cos( ), cos( )

x y a a a a

x y a a a a

y a

x a a

a a aJ

a a a

q

qq

q

q q q q q

q q q q q

-

- -

æ ö- + - -ç ÷=ç ÷+ - -è ø

æ öæ ö= - ç ÷ç ÷

+è ø è ø

- - + - +é ù= ê ú+ + +ë û

where x, y are Cartesian coordinates, 1 2,q q are joint

coordinates, 1 2,a a are link lengths, and J is the ma-

nipulator Jacobian.

We have selected polygons (square, triangle and

hexagon) in Fig. 10 in order to see improvement of

tracking performance of the proposed fuzzy controller.

The Cartesian coordinate at every sampling instant is

generated via path planning. Velocity command pro-

files of each side of the polygons are programmed to

be an LSPB (linear segments with parabolic blends)

type as shown in Fig. 8. Specifically, during 1/6 in-

terval of one side of the polygons, motion is con-

stantly accelerated, and during the next 2/3 interval of

the side, motion has constant maximum velocity, and

then during the last 1/6 interval of the side, motion is

constantly decelerated. In other words, the velocity

Fig. 8. Velocity command profile of each side of the polygons;

LSPB type between two points A and B in space. profile forms a trapezoidal shape in Cartesian space,

not in time domain.

Fig. 9 shows the whole block diagram of the con-

trol system in Cartesian space. In the figure, dx

represents the desired position vector in Cartesian

coordinate, and dθ does the desired position vector

in joint space.

Fig. 10 shows results of simulations and experi-

ments for motion tracking of square, triangle and

hexagon when the proposed variable structure fuzzy

control is applied. Tracking control performance is

evaluated by using integral of squared error (ISE) that

is defined as summation of the squared value of posi-

tion error between desired position and actual position

in Cartesian space at every sampling instant. The unit

of ISE in Fig. 10 is 2mm s× .

Fig. 10(a) is an experimental result for square path

tracking with maximum velocity 200 mm/s in the

LSPB, and Fig. 10(b) is an experimental result when

maximum velocity is doubled to be 400 mm/s. In this

case, maximum angular speeds of the motors are

within 20 % of the maximum angular speeds that can

be generated by the motors. From a comparison of (a)

and (b), we see that the ISE value is increased as

tracking speed is increased. Fig. 10(c) is the simula-

tion result at the same operating condition with (b).

Fig. 10(b) and (c) show similar tracking patterns. Fig.

10(d) is the magnified experimental result at the cor-

ner of the square of (a). Fig. 10(d) shows significant

improvement of the proposed fuzzy controller for

corner tracking compared to the conventional fuzzy

controller with two input variables.

Fig. 10(e) and (f) are experimental results to show a

comparison of control performance of the proposed

fuzzy control and classical PID control. We tuned

PID control gains roughly via Ziegler-Nichols tuning-

rules and then finely tuned. However, PID control

gains may possibly be more tuned to have less ISE

value, but it takes time and is a tedious job.

Fig. 10(g) is an experimental result when impact

force is exerted on the side of the upper link during

1362 C.-G. Kang / Journal of Mechanical Science and Technology 23 (2009) 1354~1364

operation, in order to see the response for external

disturbance. We hit the link with a thick book. The

experimental result (g) shows that the stability of the

system with the proposed fuzzy control logic is main-

tained for external disturbances with relatively big

magnitudes. This robust stability comes, we believe,

from the fact that designing the proposed fuzzy con-

troller does not depend on precise knowledge of ac-

tual dynamics and uncertainties of the system.

5. Conclusions

When a conventional fuzzy PD controller with two

input variables (position error and velocity error) is

applied to motion tracking control of mechanical

systems with complex dynamics and nonlinearities,

position errors usually remain at steady state mainly

due to friction. To reduce these errors, we have pro-

posed a revised fuzzy controller with variable struc-

ture between fuzzy PD control and fuzzy PID control

and with a switching logic between two fuzzy con-

trollers according to tracking conditions (transient or

steady state), and applied it to motion control of a

mechanical system in Cartesian space.

When the proposed variable structure fuzzy control

is applied to two DOF mechanical systems with com-

plex dynamics and nonlinearities as shown in Fig. 4,

the position errors at steady state have been decreased

more than 80 % (in magnitude) compared to ones of

the conventional fuzzy control. Moreover, the pro-

posed fuzzy control logic has improved tracking con-

trol performance significantly, especially, at corner

trackings of polygons in Cartesian space.

With the proposed implementation method of the

Fig. 9. The whole block diagram of the control system in Cartesian space.

Proposed fuzzy control

conventional

fuzzy control

(a) ISE=56 (b) ISE=77 (c) ISE=42 (d)

(e) ISE =62 (f) ISE=418 (g) Fig. 10. Position tracking for polygon paths. (a) experiment with maximum velocity = 200 mm/s, (b) experiment with maximum veloc-

ity = 400 mm/s, (c) simulation with maximum velocity = 400 mm/s, (d) comparison of the conventional fuzzy control and the proposed fuzzy control, (e) experiment with the proposed fuzzy controller, (f) experiment with PID controller, (g) experiment with impact dis-

turbances.